Rigorous validity of the Boltzmann equation for a thin layer of a rarefied gas

We consider a thin layer of a rarified gas modeled by a large hard-sphere system and show that, as long as the thickness of the layer is much larger than the interaction length, the limiting behavior is described, at least for short times, by a Boltzmann equation with two-dimensional position variable and three-dimensional velocity. By the analysis of the Lorentz gas we argue that, if the thickness of the layer is of the same order of the interaction length, this is not the case.